More about equality relation. Equality relation is defined as
antisymmetric equivalence relation. Unfortunately, the definition is
circular because antysymmetry is defined via equality! So let's focus
on equivalence relation axioms, first. We have

x=x

which can't be interpeted in any meninful way in the relational
theory. Then we have

x=y implies y=x

which isn't again a meaningful statement in the relational theory,
because any binary relation is symmetric in a sense that the order of
attributes doesn't really matter.

This can be contrasted to Tarski relation algebra where both
properties can be expressed.

Transitivity, however can be expressed. We can use set join /=\ or
natural join /\ followed by projection in order to exclude "middle"
column. So we have

`x=y` /=\ `y=z` = `x=z`

This again defines equivalence only, not equality. In order to define
equality transitivity has to be strengthened into the following law:

R(x,y) /=\ `y=z` = R(x,z)

Again, attribute agnostic expression would require to use relation R
with the same set of attributes. We therefore join the both sides of
the above identity with equality `y=z` to get

R(x,y) /=\ `y=z` /=\ `y=z` = R(x,y)

Notice that atrributes of the relation R and `y=z` "overlap" in a
certain way. Clearly the identity